EBK NUMERICAL METHODS FOR ENGINEERS
EBK NUMERICAL METHODS FOR ENGINEERS
7th Edition
ISBN: 8220100254147
Author: Chapra
Publisher: MCG
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Chapter 9, Problem 2P

A number of matrices are defined as

[ A ] = [ 4 1 5 7 2 6 ] [ B ] = [ 4 3 7 1 2 7 2 0 4 ] [ C ] = { 3 6 1 } [ D ] = 9 4 2 1 3 6 7 5

[ E ] = [ 1 5 8 7 2 3 4 0 6 ] [ F ] = [ 3 1 0 7 1 3 ] [ G ] = [ 7 6 4 ]

Answer the following questions regarding these matrices:

(a) What are the dimensions of the matrices?

(b) Identify the square, column, and row matrices.

(c) What are the values of the elements: a 12 , b 23 , d 32 , e 22 , f 12 , g 12 ?

(d) Perform the following operations:

( 1 ) [ E ] + [ B ] ( 2 ) [ A ] × ( F ) ( 3 ) [ B ] [ E ] ( 4 ) 7 × [ B ] ( 5 ) [ E ] × [ B ] ( 6 ) { C } T ( 7 ) [ B ] × [ A ] ( 8 ) [ D ] T

( 9 ) [ A ] × { C } ( 11 ) [ E ] T [ E ] ( 10 ) [ l ] × [ B ] ( 12 ) { C } T { C }

(a)

Expert Solution
Check Mark
To determine

To calculate: The dimensions of the matrices,

[A]=[471256], [B]=[437127204], [C]={361}, [D]=[94362175], [E]=[158723406], [F]=[301173], [G]=[764].

Answer to Problem 2P

Solution:

Dimension of the matrix [A] is 3×2, dimension of the matrix [B] is 3×3, dimension of the matrix [C] is 3×1, dimension of the matrix [D] is 2×4, dimension of the matrix [E] is 3×3, dimension of the matrix [F] is 2×3 and dimension of the matrix [G] is 1×3.

Explanation of Solution

Given:

The matrices,

[A]=[471256], [B]=[437127204], [C]={361}, [D]=[94362175], [E]=[158723406], [F]=[301173], [G]=[764].

Formula used:

The representation of dimension of any matrix can be written as (number of rows × number of columns).

Calculation:

Consider the matrix A,

[A]=[471256]

Matrix A has 3 rows and 2 columns.

Therefore, the dimension of the matrix A is 3×2.

Consider the matrix B,

[B]=[437127204]

Matrix B has 3 rows and 3 columns.

Therefore, the dimension of the matrix B is 3×3.

Consider the matrix C,

[C]=[361]

Matrix C has 3 rows and 1 column.

Therefore, the dimension of the matrix C is 3×1.

Consider the matrix D,

[D]=[94362175]

Matrix D has 2 rows and 4 columns.

Therefore, the dimension of the matrix D is 2×4.

Consider the matrix E,

[E]=[158723406]

Matrix E has 3 rows and 3 columns.

Therefore, the dimension of the matrix E is 3×3.

Consider the matrix F,

[F]=[301173]

The matrix F has 2 rows and 3 columns.

Therefore, the dimension of the matrix F is 2×3.

Consider the matrix G,

[G]=[764]

Matrix G has 1 row and 3 columns.

Therefore, the dimension of the matrix G is 1×3.

(b)

Expert Solution
Check Mark
To determine

The square matrix, column matrix and row matrix from the following matrices,

[A]=[471256], [B]=[437127204], [C]={361}, [D]=[94362175], [E]=[158723406], [F]=[301173], [G]=[764].

Answer to Problem 2P

Solution:

Matrices B and E are square matrices, matrix C is a column matrix and matrix G is a row matrix.

Explanation of Solution

Given:

The matrices,

[A]=[471256], [B]=[437127204], [C]={361}, [D]=[94362175], [E]=[158723406], [F]=[301173], [G]=[764].

Square matrix: A matrix M is said to be square matrix, if number of rows equals to number of columns.

Column Matrix: A matrix M is said to be column matrix, if number of rows can be any natural number but number of columns should be 1.

Row matrix: A matrix M is said to be row matrix, if number of rows should be 1 but number of columns can be any natural number.

Since, the dimension of the matrix B is 3×3.

So, number of rows equal to number of columns.

Hence, [B] is a square matrix.

Since, the dimension of the matrix C is 3×1.

So, number of rows is 3 and number of columns is 1.

Hence, [C] is a column matrix.

Since, the dimension of the matrix E is 3×3.

So, number of rows equal to number of columns.

Hence, [E] is a square matrix.

Since, the dimension of the matrix G is 1×3.

So, number of rows is 1 and number of columns is 3.

Hence, [G] is a row matrix.

Therefore, matrices B and E are square matrices, matrix C is a column matrix and matrix G is a row matrix.

(c)

Expert Solution
Check Mark
To determine

To calculate: The value of the elements a12, b23, d32, e22, f12 and g12 from the following matrices,

[A]=[471256], [B]=[437127204], [C]={361}, [D]=[94362175], [E]=[158723406], [F]=[301173], [G]=[764].

Answer to Problem 2P

Solution:

The value of the elements a12=7, b23=7, d32= does not exist, e22=2, f12=0 and g12=6.

Explanation of Solution

Given:

The matrices,

[A]=[471256], [B]=[437127204], [C]={361}, [D]=[94362175], [E]=[158723406], [F]=[301173], [G]=[764].

Formula used:

The value of the element aij is the (i,j)th value of the matrix, where (i,j) denote the ith row and jth column of the matrix.

Calculation:

Consider the matrix A,

[A]=[471256]

Then the value of a12 is the (1,2)th value of the matrix A, that is 7.

Therefore, a12=7.

Consider the matrix B,

[B]=[437127204]

Then the value of b23 is the (2,3)th value of the matrix B, that is 7.

Therefore, b23=7.

Consider the matrix D,

[D]=[94362175]

Then the value of d32 is the (3,2)th value of the matrix D.

Since, matrix D does not have 3rd row. Hence, d32 does not exist.

Therefore, d32= does not exist.

Consider the matrix E,

[E]=[158723406]

Then the value of e22 is the (2,2)th value of the matrix E, that is 2.

Therefore, e22=2.

Consider the matrix F,

[F]=[301173]

Then the value of f12 is the (1,2)th value of the matrix F, that is 0.

Therefore, f12=0.

Consider the matrix G,

[G]=[764]

Then the value of g12 is the (1,2)th value of the matrix G, that is 6.

Therefore, g12=6.

(d)

Expert Solution
Check Mark
To determine

To calculate: The following operations,

(1) [E]+[B],

(2) [A]×[F],

(3) [B][E],

(4) 7×[B],

(5) [E]×[B],

(6) {C}T,

(7) [B]×[A],

(8) [D]T,

(9) [A]×{C},

(10) [I]×[B],

(11) [E]T×[E] and

(12) {C}T×{C}.

Where,

[A]=[471256], [B]=[437127204], [C]={361}, [D]=[94362175], [E]=[158723406], [F]=[301173], [G]=[764].

Answer to Problem 2P

Solution:

(1) [E]+[B]=[581584106010], (2) [A]×[F]=[1949255147214223], (3) [B][E]=[321604202],

(4) 7×[B]=[2821497144914028], (5) [E]×[B]=[251374362575281252], (6) {C}T={361},

(7) [B]×[A]=[547641532838], (8) [D]T=[92413765], (9) [A]×{C}= not possible,

(10) [I]×[B]=[B], (11) [E]T×[E]=[6619531929465346109], (12) {C}T×{C}=[46].

Explanation of Solution

Given:

The matrices,

[A]=[471256], [B]=[437127204], [C]={361}, [D]=[94362175], [E]=[158723406], [F]=[301173], [G]=[764].

Formula used:

If A=[abcd] and B=[efgh] be any two matrices.

Addition of A and B is denoted by A+B and equal to [a+eb+fc+gd+h].

Subtraction of A and B is denoted by AB and equal to [aebfcgdh].

Multiply any scalar quantity 5 in the matrix A is denoted by 5A and equal to [5a5b5c5d].

Transpose of the matrix A is denoted by AT and equal to [acbd].

Multiplication of the matrices A and B is denoted by [A×B] and equal to [a.e+b.ga.f+b.hc.e+d.gc.f+d.h].

Multiplication of two matrices is possible if interior dimensions are equal.

If matrix A has dimension m×n and matrix B has dimension p×q. Then [A]×[B] is possible only if n=p. Interior dimensions of [A]×[B] is n and p. Exterior dimensions of [A]×[B] is m and q.

Calculation:

(1) Consider the matrices E and B,

[E]=[158723406] and [B]=[437127204].

Now, the addition of two matrices is,

[E]+[B]=[158723406]+[437127204]=[1+45+38+77+12+23+74+20+06+4]=[581584106010]

Therefore, [E]+[B]=[581584106010].

(2) Consider the matrices A and F,

[A]=[471256] and [F]=[301173].

Now, the multiplication of two matrices is,

[A]×[F]=[471256]×[301173]=[4×3+7×14×0+7×74×1+7×31×3+2×11×0+2×71×1+2×35×3+6×15×0+6×75×1+6×3]=[1949255147214223]

Therefore, [A]×[F]=[1949255147214223].

(3) Consider the matrices B and E,

[B]=[437127204] and [E]=[158723406].

Now, the subtraction of two matrices is,

[B][E]=[437127204][158723406]=[413578172273240046]=[321604202]

Therefore, [B][E]=[321604202].

(4) Consider a matrix B, and a scalar 7,

[B]=[437127204].

The multiplication of a matrix B with a scalar 7 is,

7×[B]=7×[437127204]=[7×47×37×77×17×27×77×27×07×4]=[2821497144914028]

Therefore, 7×[B]=[2821497144914028].

(5) Consider the matrices E and B,

[E]=[158723406] and [B]=[437127204].

Now the multiplication of two matrices E and B is,

[E]×[B]=[158723406]×[437127204]=[1×4+5×1+8×21×3+5×2+8×01×7+5×7+8×47×4+2×1+3×27×3+2×2+3×07×7+2×7+3×44×4+0×1+6×24×3+0×2+6×04×7+0×7+6×4]=[251374362575221252]

Therefore, [E]×[B]=[251374362575221252].

(6) Consider the matrix C,

[C]={361}

Now the transpose of the matrix C is,

[C]T={361}

Therefore, [C]T={361}.

(7) Consider the matrices B and A,

[B]=[437127204] and [A]=[471256]

Now the multiplication of the two matrices B and A is,

[B]×[A]=[437127204]×[471256]=[4×4+3×1+7×54×7+3×2+7×61×4+2×1+7×51×7+2×2+7×62×4+0×1+4×52×7+0×2+4×6]=[547641532838]

Therefore, [B]×[A]=[547641532838]

(8) Consider the matrix D,

[D]=[94362175]

Now, the transpose of the matrix D is,

[D]T==[92413765].

Therefore, [D]T==[92413765].

(9) Consider the matrices A and C,

[A]=[471256] and [C]={361}.

Since, interior dimensions of [A]×{C} are not equals. Hence, [A]×{C} is not possible.

Therefore, [A]×{C} is not possible.

(10) Consider the identity matrix I of dimension 3×3 and matrix B,

[B]=[437127204].

Since, the multiplication of the identity matrix with any matrix is again that matrix.

Hence,

[I]×[B]=[100010001]×[437127204]=[437127204]

Therefore, [I]×[B]=[B].

(11) Consider the matrix E,

[E]=[158723406].

Now the transpose of the matrix E is,

[E]T=[174520836].

The multiplication of the matrix E and its transpose is,

[E]T×[E]=[174520836]×[158723406]=[1×1+7×7+4×41×5+7×2+4×01×8+7×3+4×65×1+2×7+0×45×5+2×2+0×05×8+2×3+0×68×1+3×7+6×48×5+3×2+6×08×8+3×3+6×6]=[6619531929465346109]

Therefore, [E]T×[E]=[6619531929465346109].

(12) Consider the matrix C,

[C]={361}

Now the transpose of the matrix C is,

[C]T={361}

Multiplication of the matrix C and its transpose is,

[C]T×[C]={361}×{361}=[3×3+6×6+1×1]=[46]

Therefore, [C]T×[C]=[46].

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